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A Note on Groups Associated with 4‐Arc‐Transitive Cubic Graphs
Author(s) -
Conder Marston,
Morton Margaret
Publication year - 1990
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/22.6.553
Subject(s) - mathematics , transitive relation , arc (geometry) , cubic graph , combinatorics , graph , line graph , geometry , voltage graph
A cubic (trivalent) graph Γ is said to be 4‐arc‐transitive if its automorphism group acts transitively on the 4‐arcs of Γ (where a 4‐arc is a sequence v 0 , v 1 , … v 4 of vertices of Γ such that v i −1 is adjacent to v i for 1 ⩽ i ⩽ 4, and v i −1 ≠ v i +1 for 1 ⩽ i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4 + ( a m ), for m = 3,4,5…, each presented in terms of generators and relations under the additional assumption that the vertices of a circuit of length m are cyclically permuted by some automorphism. In this paper it is shown that whenever m is a proper multiple of 6, the group 4 + ( a m ) is infinite. The proof is obtained by constructing transitive permutation representations of arbitrarily large degree.